Trignometrical Equations Test 1

Total Questions:50 Total Time: 60 Min

Remaining:

 

Questions 1 of 50

Question:If \({\sin ^2}\theta = \frac{1}{4},\)then the most general value of \(\theta \)is

Answers Choices:

\(2n\pi \pm {( - 1)^n}\frac{\pi }{6}\)

\(\frac{{n\pi }}{2} \pm {( - 1)^n}\frac{\pi }{6}\)

\(n\pi \pm \frac{\pi }{6}\)

\(2n\pi \pm \frac{\pi }{6}\)

Questions 2 of 50

Question:If \({\sec ^2}\theta = \frac{4}{3}\), then the general value of \(\theta \)is

Answers Choices:

\(2n\pi \pm \frac{\pi }{6}\)

\(n\pi \pm \frac{\pi }{6}\)

\(2n\pi \pm \frac{\pi }{3}\)

\(n\pi \pm \frac{\pi }{3}\)

Questions 3 of 50

Question:General solution of the equation \(\cot \theta - \tan \theta = 2\)is

Answers Choices:

\(n\pi + \frac{\pi }{4}\)

\(\frac{{n\pi }}{2} + \frac{\pi }{8}\)

\(\frac{{n\pi }}{2} \pm \frac{\pi }{8}\)

None of these

Questions 4 of 50

Question:If \(\cot \theta + \tan \theta = 2{\rm{cosec}}\theta \), the general value of \(\theta \) is

Answers Choices:

\(n\pi \pm \frac{\pi }{3}\)

\(n\pi \pm \frac{\pi }{6}\)

\(2n\pi \pm \frac{\pi }{3}\)

\(2n\pi \pm \frac{\pi }{6}\)

Questions 5 of 50

Question:If \({\tan ^2}\theta - (1 + \sqrt 3 )\tan \theta + \sqrt 3 = 0\), then the general value of \(\theta \) is

Answers Choices:

\(n\pi + \frac{\pi }{4},n\pi + \frac{\pi }{3}\)

\(n\pi - \frac{\pi }{4},n\pi + \frac{\pi }{3}\)

\(n\pi + \frac{\pi }{4},n\pi - \frac{\pi }{3}\)

\(n\pi - \frac{\pi }{4},n\pi - \frac{\pi }{3}\)

Questions 6 of 50

Question:The most general value of \(\theta \)satisfying the equations \(\sin \theta = \sin \alpha \)and \(\cos \theta = \cos \alpha \)is

Answers Choices:

\(2n\pi + \alpha \)

\(2n\pi - \alpha \)

\(n\pi + \alpha \)

\(n\pi - \alpha \)

Questions 7 of 50

Question:The solution of the equation \(4{\cos ^2}x + 6\)\({\sin ^2}x = 5\)

Answers Choices:

\(x = n\pi \pm \frac{\pi }{2}\)

\(x = n\pi \pm \frac{\pi }{4}\)

\(x = n\pi \pm \frac{{3\pi }}{2}\)

None of these

Questions 8 of 50

Question:If \(\sin 3\alpha = 4\sin \alpha \sin (x + \alpha )\sin (x - \alpha ),\)then \(x = \)

Answers Choices:

\(n\pi \pm \frac{\pi }{6}\)

\(n\pi \pm \frac{\pi }{3}\)

\(n\pi \pm \frac{\pi }{4}\)

\(n\pi \pm \frac{\pi }{2}\)

Questions 9 of 50

Question:If \(\sin {\rm{ }}\left( {\frac{\pi }{4}\cot \theta } \right) = \cos {\rm{ }}\left( {\frac{\pi }{4}\tan \theta } \right)\,\,,\) then \(\theta = \)

Answers Choices:

\(n\pi + \frac{\pi }{4}\)

\(2n\pi \pm \frac{\pi }{4}\)

\(n\pi - \frac{\pi }{4}\)

\(2n\pi \pm \frac{\pi }{6}\)

Questions 10 of 50

Question:If \(\sin 2\theta = \cos 3\theta \)and \(\theta \)is an acute angle, then \(\sin \theta \)is equal to [

Answers Choices:

\(\frac{{\sqrt 5 - 1}}{4}\)

\(\frac{{ - \sqrt 5 - 1}}{4}\)

0

None of these

Questions 11 of 50

Question:If \(4{\sin ^4}x + {\cos ^4}x = 1,\)then x =

Answers Choices:

\(n\pi \)

\(n\pi \pm {\sin ^{ - 1}}\frac{2}{5}\)

\(n\pi + \frac{\pi }{6}\)

None of these

Questions 12 of 50

Question:If \(\cos 3x + \sin \left( {2x - \frac{{7\pi }}{6}} \right) = - 2\), then \(x = \) (where \(k \in Z\))

Answers Choices:

\(\frac{\pi }{3}(6k + 1)\)

\(\frac{\pi }{3}(6k + 1)\)

\(\frac{\pi }{3}(2k + 1)\)

None of these

Questions 13 of 50

Question:The equation \({\sin ^4}x + {\cos ^4}x + \sin 2x + \alpha = 0\)is solvable for

Answers Choices:

\( - \frac{1}{2} \le \alpha \le \frac{1}{2}\)

\( - 3 \le \alpha \le 1\)

\( - \frac{3}{2} \le \alpha \le \frac{1}{2}\)

\( - 1 \le \alpha \le 1\)

Questions 14 of 50

Question:If \(|k|\, = 5\)and \({0^o} \le \theta \le {360^o}\), then the number of different solutions of 3\(\cos \theta + 4\sin \theta = k\) is

Answers Choices:

Zero

Two

One

Infinite

Questions 15 of 50

Question:If \(0 \le x \le \pi \)and \({81^{{{\sin }^2}x}} + {81^{{{\cos }^2}x}} = 30\), then x =

Answers Choices:

\(\pi /6\)

\(\pi /2\)

\(\pi /4\)

\(3\pi /4\)

Questions 16 of 50

Question:If \(\sin \theta = \sqrt 3 \cos \theta , - \pi < \theta < 0\), then \(\theta = \)

Answers Choices:

\( - \frac{{5\pi }}{6}\)

\( - \frac{{4\pi }}{6}\)\(\)

\(\frac{{4\pi }}{6}\)

\(\frac{{5\pi }}{6}\)

Questions 17 of 50

Question:The only value of x for which \({2^{\sin x}} + {2^{\cos x}} > {2^{1 - (1/\sqrt 2 )}}\) holds, is

Answers Choices:

\(\frac{{5\pi }}{4}\)

\(\frac{{3\pi }}{4}\)

\(\frac{\pi }{2}\)

All values of x

Questions 18 of 50

Question:If \((1 + \tan \theta )(1 + \tan \varphi ) = 2\), then \(\theta + \varphi \)=

Answers Choices:

\({30^o}\)

\({45^o}\)

\({60^o}\)

\({75^o}\)

Questions 19 of 50

Question:Period of \(|2\sin 3\theta + 4\cos 3\theta |\)is

Answers Choices:

\(\frac{{2\pi }}{3}\)

\(\pi \)

\(\frac{\pi }{2}\)

\(\frac{\pi }{3}\)

Questions 20 of 50

Question:The period of \({\sin ^4}x + {\cos ^4}x\)is

Answers Choices:

\(\pi /2\)

\(\pi \)

\(2\pi \)

\(3\pi /2\)

Questions 21 of 50

Question:If the angles of a triangle \(ABC\)be in A.P., then

Answers Choices:

\({c^2} = {a^2} + {b^2} - ab\)

\({b^2} = {a^2} + {c^2} - ac\)

\({a^2} = {b^2} + {c^2} - ac\)

\({b^2} = {a^2} + {c^2}\)

Questions 22 of 50

Question:In triangle \(ABC\), \((b + c)\cos A + (c + a)\cos B\) \( + (a + b)\cos C = \)

Answers Choices:

0

1

\(a + b + c\)

\(2(a + b + c)\)

Questions 23 of 50

Question:\(\angle A + \angle C = {90^o}\)In\(\Delta ABC,\) if \(2(bc\cos A + ca\cos B + ab\cos C) = \)

Answers Choices:

0

\(a + b + c\)

\({a^2} + {b^2} + {c^2}\)

None of these

Questions 24 of 50

Question:In \(\Delta ABC,\) \({\rm{cosec }}A(\sin B\cos C + \cos B\sin C) = \)

Answers Choices:

\(c/a\)

\(a/c\)

1

\(c/ab\)

Questions 25 of 50

Question:In \(\Delta ABC\), \(c\cos (A - \alpha ) + a\cos (C + \alpha ) = \)

Answers Choices:

\(a\cos \alpha \)

\(b\cos \alpha \)

\(c\cos \alpha \)

\(2b\cos \alpha \)

Questions 26 of 50

Question:In \(\Delta ABC\), \(\frac{{\cos A}}{a} + \frac{{\cos B}}{b} + \frac{{\cos C}}{c} = \)

Answers Choices:

\(\frac{{{a^2} + {b^2} + {c^2}}}{{abc}}\)

\(\frac{{{a^2} + {b^2} + {c^2}}}{{2abc}}\)

\(\frac{{2({a^2} + {b^2} + {c^2})}}{{abc}}\)

\({a^2} + {b^2} + {c^2}\)

Questions 27 of 50

Question:If the angles \(A,B,C\)of a triangle are in A.P. and the sides \(a,b,c\) opposite to these angles are in G. P. then \({a^2},{b^2},{c^2}\) are in

Answers Choices:

A. P.

H. P.

G. P.

None of these

Questions 28 of 50

Question:If the sides of a triangle are p,q and\(\sqrt {{p^2} + pq + {q^2}} \), then the biggest angle is

Answers Choices:

\(\pi /2\)

\(2\pi /3\)

\(5\pi /4\)

\(7\pi /4\)

\(5\pi /3\)

Questions 29 of 50

Question:The perimeter of a \(\Delta ABC\)is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

Answers Choices:

\(\frac{\pi }{6}\)

\(\frac{\pi }{3}\)

\(\frac{\pi }{2}\)

\(\pi \)

Questions 30 of 50

Question:Point D, E are taken on the side BC of a triangle \(ABC\)such that \(BD = DE = EC\).If \(\angle BAD = x\), \(\angle DAE = y\), \(\angle EAC = z\), then the value of \(\frac{{\sin (x + y)\sin (y + z)}}{{\sin x\sin z}} = \)

Answers Choices:

1

2

4

None of these

Questions 31 of 50

Question:If in a triangle ABC side \(a = (\sqrt 3 + 1)\)cms and \(\angle B = {30^o},\) \(\angle C = {45^o}\), then the area of the triangle is

Answers Choices:

\(\frac{{\sqrt 3 + 1}}{3}c{m^2}\)

\(\frac{{\sqrt 3 + 1}}{2}c{m^2}\)

\(\frac{{\sqrt 3 + 1}}{{2\sqrt 2 }}c{m^2}\)

\(\frac{{\sqrt 3 + 1}}{{3\sqrt 2 }}c{m^2}\)

Questions 32 of 50

Question:If in a right angled triangle the hypotenuse is four times as long as the perpendicular drawn to it from opposite vertex, then one of its acute angle is

Answers Choices:

\({15^o}\)

\({30^o}\)

\({45^o}\)

None of these

Questions 33 of 50

Question:In a\(\Delta ABC,\)if \(b = 20,c = 21\)and \(\sin A = 3/5\), then \(a = \)

Answers Choices:

12

13

14

15

Questions 34 of 50

Question:Let D be the middle point of the side BC of a triangle ABC. If the triangle ADC is equilateral, then \({a^2}:{b^2}:{c^2}\)is equal to

Answers Choices:

\(1:4:3\)

\(4:1:3\)

\(4:3:1\)

\(3:4:1\)

Questions 35 of 50

Question:If the sides of triangle be 6, 10 and 14 then the triangle is

Answers Choices:

Obtuse angled

Acute angled

Right angled

Equilateral

Questions 36 of 50

Question:In any \(\Delta ABC\)if \(a\cos B = b\cos A\), then the triangle is

Answers Choices:

Equilateral triangle

Isosceles triangle

Scalene

Right angled

Questions 37 of 50

Question:If A is the area and 2s the sum of 3 sides of triangle, then

Answers Choices:

\(A \le \frac{{{s^2}}}{{3\sqrt 3 }}\)

\(A \le \frac{{{s^2}}}{2}\)

\(A > \frac{{{s^2}}}{{\sqrt 3 }}\)

None of these

Questions 38 of 50

Question:If in a triangle \(ABC\)right angled at \(B,s - a = 3\), \(s - c = 2\), then the values of a and c are respectively

Answers Choices:

2, 3

3, 4

4, 3

6, 8

Questions 39 of 50

Question:The area of triangle \(ABC,\) in which \(a = 1,\;b = 2\), \(\angle C = 60^\circ \)is

Answers Choices:

\(\frac{1}{2}\)

\(\sqrt 3 \)

\(\frac{{\sqrt 3 }}{2}\)

\(\frac{3}{2}\)

Questions 40 of 50

Question:In a triangle \(ABC\), if \(b + c = 2a\) and \(\angle A = 60^\circ ,\) then \(\Delta ABC\) is

Answers Choices:

Scalene

Equilateral

Isosecles

Right angled

Questions 41 of 50

Question:In any triangle ABC, \(a\cot A + b\cot B + c\cot C = \)

Answers Choices:

\(r + R\)

\(r - R\)

\(2(r + R)\)

\(2(r - R)\)

Questions 42 of 50

Question:If the radius of the circumcircle of an isosceles triangle \(PQR\) is equal to \(PQ( = PR),\)then the angle P is

Answers Choices:

\(\frac{\pi }{6}\)

\(\frac{\pi }{3}\)

\(\frac{\pi }{2}\)

\(\frac{{2\pi }}{3}\)

Questions 43 of 50

Question:The angle of elevation of a tower at a point distant d meters from its base is\({\rm{3}}0^\circ \). If the tower is 20 meters high, then the value of d is

Answers Choices:

\(10\sqrt 3 m\)

\(\frac{{20}}{{\sqrt 3 }}m\)

\(20\sqrt 3 m\)

\(10\)m

Questions 44 of 50

Question:The angle of elevation of the top of the tower observed from each of the three points \(A,B,C\)on the ground, forming a triangle is the same angle \(\alpha \). If R is the circum-radius of the triangle ABC, then the height of the tower is

Answers Choices:

\(R\sin \alpha \)

\(R\cos \alpha \)

\(R\cot \alpha \)

\(R\tan \alpha \)

Questions 45 of 50

Question:From a point a metre above a lake the angle of elevation of a cloud is a and the angle of depression of its reflection is \(\beta \). The height of the cloud is

Answers Choices:

\(\frac{{a\sin \,(\alpha + \beta )}}{{\sin \,(\alpha - \beta )}}\)

metre\(\frac{{a\sin \,(\alpha + \beta )}}{{\sin \,(\beta - \alpha )}}\) metre

\(\frac{{a\sin \,(\beta - \alpha )}}{{\sin \,(\alpha + \beta )}}\) Metre

None of these

Questions 46 of 50

Question:If the angle of depression of a point A on the ground from the top of a tower be \(30^\circ \), then the angle of elevation of the top of the tower from the point A will be

Answers Choices:

\(60^\circ \)

\(45^\circ \)

\(30^\circ \)

None of these

Questions 47 of 50

Question:The angle of depression of a ship from the top of a tower 30 metre high is \({60^0}\), then the distance of ship from the base of tower is [MP PET 1988; Pb. CET 2003]

Answers Choices:

30 m

\(30\,\sqrt 3 \,\,m\)

\(10\sqrt 3 \,m\)

10 m

Questions 48 of 50

Question:The angle of elevation of a stationary cloud from a point 2500 m above a lake is \({15^0}\) and the angle of depression of its reflection in the lake is \({45^0}\). The height of cloud above the lake level is

Answers Choices:

\(2500\,\sqrt 3 \,metres\)

2500 metres

\(500\,\sqrt 3 \,metres\)

None of these

Questions 49 of 50

Question:A person is standing on a tower of height \(15(\sqrt 3 + 1)\,m\) and observing a car coming towards the tower. He observed that angle of depression changes from \({30^0}\) to \({45^0}\) in 3 sec. What is the speed of the car

Answers Choices:

36 km/hr

72 km/hr

18 km/hr

30 km/hr

Questions 50 of 50

Question:Two men are on the opposite side of a tower. They measure the angles of elevation of the top of the tower \({45^0}\) and \({30^0}\) respectively. If the height of the tower is 40 m, find the distance between the men

Answers Choices:

40 m

\(40\sqrt 3 \,m\)

68.280 m

109.28 m